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Π
π '''(pi) is the 16th letter of the Greek alphabet. The radius is actually counted as the number 3.14159: There are more numbers behind 3.14159: ''3.14159 26535 89793 23846 26433...'' They are 5 (or more) groups in the symbol pi. Name The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter π, sometimes spelled out as pi, and derived from the first letter of the Greek word perimetros, meaning circumference. In English, π is pronounced as "pie" (/paɪ/ PY). In mathematical use, the lowercase letter π (or π in sans-serif font) is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. Definition π is commonly defined as the ratio of a circle's circumference C'' to its diameter ''d: The ratio C''/''d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C''/''d. This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C''/''d. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits, a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x''2 + ''y''2 = 1, as the integral: An integral such as this was adopted as the definition of π by Karl Weierstrass, who defined it directly as an integral in 1841. Definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. Remmert (1991) explains that this is because in many modern treatments of calculus, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following: π is twice the smallest positive number at which the cosine function equals 0. The cosine can be defined independently of geometry as a power series, or as the solution of a differential equation. In a similar spirit, π can be defined instead using properties of the complex exponential, exp ''z, of a complex variable z''. Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which exp ''z is equal to one is then an (imaginary) arithmetic progression of the form: and there is a unique positive real number π with this property. A more abstract variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem: there is a unique (up to automorphism) continuous isomorphism from the group '''R/'Z' of real numbers under addition modulo integers (the circle group) onto the multiplicative group of complex numbers of absolute value one. The number π is then defined as half the magnitude of the derivative of this homomorphism. A circle encloses the largest area that can be attained within a given perimeter. Thus the number π is also characterized as the best constant in the isoperimetric inequality (times one-fourth). There are many other, closely related, ways in which π appears as an eigenvalue of some geometrical or physical process; see below. Irrationality and normality π is an irrational number, meaning that it cannot be written as the ratio of two integers (fractions such as 22/7 and 355/113 are commonly used to approximate π, but no common fraction (ratio of whole numbers) can be its exact value). Because π is irrational, it has an infinite number of digits in its decimal representation, and it does not settle into an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e'' or ln 2 but smaller than the measure of Liouville numbers. The digits of π have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that π is normal has not been proven or disproven. Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found. Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem. Thus, because the sequence of π's digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π. This is also called the "Feynman point" in mathematical folklore, after Richard Feynman, although no connection to Feynman is known. Transcendence In addition to being irrational, more strongly π is a transcendental number, which means that it is not the solution of any non-constant polynomial equation with rational coefficients, such as The transcendence of π has two important consequences: First, π cannot be expressed using any finite combination of rational numbers and square roots or ''n-th roots such as 3√31 or √10. Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle. Squaring a circle was one of the important geometry problems of the classical antiquity. Amateur mathematicians in modern times have sometimes attempted to square the circle and sometimes claim success despite the fact that it is mathematically impossible. Continued fractions Like all irrational numbers, π cannot be represented as a common fraction (also known as a simple or vulgar fraction), by the very definition of "irrational number" (that is, "not a rational number"). But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction: Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3, 22/7, 333/106, and 355/113. These numbers are among the most well-known and widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator. Because π is known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, π cannot have a periodic continued fraction. Although the simple continued fraction for π (shown above) also does not exhibit any other obvious pattern, mathematicians have discovered several generalized continued fractions that do. Approximate value and digits Some approximations of pi include: * 'Integers: '''3 * '''Fractions: '''Approximate fractions include (in order of increasing accuracy) 22/7, 333/106, 355/113, 52163/16604, 103993/33102, 104348/33215, and 245850922/78256779. (List is selected terms from OEIS: A063674 and OEIS: A063673.) * '''Digits: '''The first 50 decimal digits are 3.14159265358979323846264338327950288419716939937510...(see OEIS: A000796) Digits in other number systems * The first 48 binary (base 2) digits (called bits) are 11.001001000011111101101010100010001000010110100011... (see OEIS: A004601) * The first 20 digits in hexadecimal (base 16) are 3.243F6A8885A308D31319... (see OEIS: A062964) * The first five sexagesimal (base 60) digits are 3;8,29,44,0,47 (see OEIS: A060707) Complex numbers and Euler's identity Any complex number, say ''z, can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or r'') is used to represent ''z's distance from the origin of the complex plane and the other (angle or φ) to represent a counter-clockwise rotation from the positive real line as follows: where i'' is the imaginary unit satisfying ''i''2 = −1. The frequent appearance of π in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula: where the constant ''e is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of e'' and points on the unit circle centered at the origin of the complex plane. Setting ''φ = π in Euler's formula results in Euler's identity, celebrated by mathematicians because it contains the five most important mathematical constants: There are n'' different complex numbers ''z satisfying zn = 1, and these are called the "n''-th roots of unity". They are given by this formula: History Antiquity The best-known approximations to π dating before the Common Era were accurate to two decimal places; this was improved upon in Chinese mathematics in particular by the mid-first millennium, to an accuracy of seven decimal places. After this, no further progress was made until the late medieval period. Based on the measurements of the Great Pyramid of Giza (c. 2560 BC) , some Egyptologists have claimed that the ancient Egyptians used an approximation of π as 22/7 from as early as the Old Kingdom. This claim has met with skepticism. The earliest written approximations of π are found in Babylon and Egypt, both within one percent of the true value. In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.125. In Egypt, the Rhind Papyrus, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.16. Astronomical calculations in the ''Shatapatha Brahmana (ca. 4th century BC) use a fractional approximation of 339/108 ≈ 3.139 (an accuracy of 9×10−4). Other Indian sources by about 150 BC treat π as √10 ≈ 3.1622. Polygon approximation era The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes. This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as "Archimedes' constant". Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (that is 3.1408 < π < 3.1429). Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7. Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits. In ancient China, values for π included 3.1547 (around 1 AD), √10 (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556). Around 265 AD, the Wei Kingdom mathematician Liu Hui created a polygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416. Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4. The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that 3.1415926 < π < 3.1415927 and suggested the approximations π ≈ 355/113 = 3.14159292035... and π ≈ 22/7 = 3.142857142857..., which he termed the Milü (close ratio") and ''Yuelü ("approximate ratio"), respectively, using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of remained the most accurate approximation of π available for the next 800 years. The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD). Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes. Italian author Dante apparently employed the value 3+√2/10 ≈ 3.14142. The Persian astronomer Jamshīd al-Kāshī produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3×228 sides, which stood as the world record for about 180 years. French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides. Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century). Dutch scientist Willebrord Snellius reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 1040 sides, which remains the most accurate approximation manually achieved using polygonal algorithms. Infinite series